It is very intuitive that the set $$ S=\{{x,y}\in\Bbb{R}^2\mid 2x^6+3y^4=1\} $$ is a simple closed curve.
How can one show that this is indeed true?
Does this question relate to some theorems regarding algebraic curves (I know nothing about this topic but the definition)?
Unusually easy in this case. Your curve bounds a convex body. Slightly more generally, it is star-shaped around the origin. This just means that it can be parametrized nicely in polar coordinates, given some $\theta$ there is a single positive $r$ that goes with the curve itself, points with smaller $r$ are inside the curve, larger $r$ outside.